A181293 Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's (0<=k<=n) A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.

1, 0, 2, 1, 2, 4, 2, 6, 8, 8, 4, 14, 24, 24, 16, 8, 32, 64, 80, 64, 32, 16, 72, 164, 240, 240, 160, 64, 32, 160, 408, 680, 800, 672, 384, 128, 64, 352, 992, 1848, 2480, 2464, 1792, 896, 256, 128, 768, 2368, 4864, 7296, 8288, 7168, 4608, 2048, 512, 256, 1664, 5568

Offset

0,3

Comments

The sum of entries in row n is A003480(n).

Sum_{k=0..n} k*T(n,k) = A181294(n).

Links

Table of n, a(n) for n=0..57.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Formula

G.f.: G(t,z) = (1-z)^2/(1-2*z-2*t*z+2*t*z^2).

G.f. of column k is 2^k*z^k*(1-z)^{k+2}/(1-2*z)^{k+1} (we have a Riordan array).

Example

T(2,0)=1, T(2,1)=2, T(2,2)=4 because the 2-compositions of 2, written as (top row/bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1).
Triangle starts:
1;
0,2;
1,2,4;
2,6,8,8;
4,14,24,24,16;
...

Maple

G := (1-z)^2/(1-2*z-2*t*z+2*t*z^2): Gser := simplify(series(G, z = 0, 14)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

Crossrefs

Cf. A003480, A181294.

Sequence in context: A329688 A217920 A137406 * A262876 A263045 A395351

Adjacent sequences: A181290 A181291 A181292 * A181294 A181295 A181296

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 12 2010

Status

approved